论文信息 - New bounds on the average distance from the Fermat-Weber center of a planar convex body

New bounds on the average distance from the Fermat-Weber center of a planar convex body

The Fermat-Weber center of a planar body Q is a point in the plane from which the average distance to the points in Q is minimal. We first show that for any convex body Q in the plane, the average distance from the Fermat-Weber center of Q to the points in Q is larger than [email protected][email protected](Q), where @D(Q) is the diameter of Q. This proves a conjecture of Carmi, Har-Peled and Katz. From the other direction, we prove that the same average distance is at most 2(4-3)[email protected][email protected](Q)<[email protected][email protected](Q). The new bound substantially improves the previous bound of [email protected][email protected](Q)[email protected][email protected](Q) due to Abu-Affash and Katz, and brings us closer to the conjectured value of [email protected][email protected](Q). We also confirm the upper bound conjecture for centrally symmetric planar convex bodies.

Csaba D. Tóth | Adrian Dumitrescu | Minghui Jiang | Minghui Jiang | A. Dumitrescu

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